#330824

order	512
nilpotency class	3
isoclinism	isoclinic class of #329515
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

504 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,5,6,4,7,3,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(26,30)(31,39)(32,37)(33,40);(10,14)(15,21)(16,22)(23,24)(25,26,28,30)(27,33,34,40)(29,37,36,32)(31,38,39,35);(2,6)(3,5)(7,8)(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(25,27,28,34)(26,31,30,39)(29,38,36,35)(32,40,37,33);(2,7)(6,8)(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(27,34)(31,39)(33,40)(35,38)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=e,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238020);